Instability of a Gaseous Envelope Surrounding a Planetary Core and Formation of Giant Planets

We consider the evolution of the solar nebula in stages where, as studied previously by Hayashi, Nakazawa and Adachi, protoplanets composed of involatile materials are growing gradually through the capture of planetesimals. When a protoplanet becomes greater than the moon's mass, an appreciable amount of the gas of the solar nebula is attracted by the protoplanet to form a gaseous envelope surrounding it. We have studied the structure and stability of this envelope, which depend on the mass of the protoplanet, on the assumption that the envelope is spherically-symmetric, in hydro-static equilibrium and, thermally, isothermal in the outer optically-thin region but adiabatic in the inner region. The existence of the isothermal region is due to a circumstance that the opacity of the gas is very low since almost all of grains have already condensed into planetesimals and protoplanets. We have found that, when the mass of a protoplanet becomes greater than a certain critical value which depends on the opacity, the envelope can no longer be in hydrostatic equilibrium and begins to collapse. For a roughly estimated value of the opacity, the critical mass is of the order of 15ME and 6ME (ME being the Earth's mass) for proto-Jupiter and proto-Saturn, respectively. These masses are about one fifth and one tenth of the values found by Perri and Cameron, but they are consistent with recent Slattery's models of the present Jupiter and Saturn.